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M3L12d.txt
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M3L12d.txt
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#
# File: content-mit-8-421-3x-subtitles/M3L12d.txt
#
# Captions for 8.421x module
#
# This file has 164 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
So let me summarize.
We have discussed three different ways
how we can have coupling matrix elements between two
states-- electric dipole, magnetic dipole,
electric quadrupole, E1, M1, E2.
The operator was the electric dipole operator.
Here, it was the operator of the magnetic moment,
which is orbital angular momentum and spin angular
momentum.
And for the quadrupole, it was the quadratic expression
in the spatial coordinates.
You can also ask what is the parity.
The electric dipole operator connects states
with opposite parity whereas force magnetic dipole
and electric quadrupole connects states
with even the same parity.
Magnetic dipole and electric quadrupole transition
are often called forbidden transitions.
Well, you would say it's a misnomer
because they're transitions.
So they are allowed but they are weak.
But this is the language we use.
Weak transitions are forbidden, which
simply means they don't appear.
They are forbidden in leading order.
But when you go to high order, they are allowed.
We can, of course, say if they're completely forbidden,
there would be no need to discuss them.
But since they are only forbidden
at a certain level, then of course
it's interesting to discuss them.
And a lot of narrow transitions which
are relevant for atomic clocks are highly forbidden
transactions.
The strengths of them, which usually
scales if transitions with the square of the matrix element,
is on the order of 5 times 10 to the minus 5.
So those transitions are four or five orders
of magnitude B weaker than an allowed E1 transition.
Questions.
Yes.
People talk about highly forbidden transitions.
Does this mean that it's like an optical transition,
or how do we distinguish it from just [INAUDIBLE]?
It could be.
Actually, I would say forbidden transitions are weaker
by alpha to the power N. Here, we
have a situation where the matrix element
is just smaller by alpha.
But yes, you could have-- sometimes, yes.
Some transitions are highly forbidden.
For instance, I try to remember.
If you have hydrogen 1S and 2S-- because S states-- actually,
you're asking questions about the next chapter,
mainly about selection holds.
If you have-- but get inverts.
The S, if you connect two S states,
they have both zero angular momentum.
So you cannot have a quadrupole operator connecting the two.
It would violate the triangle rule.
You cannot have angular momentum of zero and angular momentum
of two and get angular momentum of zero.
So therefore, you have a transition between two states
of the same parity.
There is no dipole operator.
There is no quadrupole operator.
So you soon run into a situation where it's highly forbidden.
Sometimes, you have the situation
that something is forbidden in non-relativistic physics,
but the relativistic term which makes it allowed.
So then it's not-- well, this also, we
know that relativistic terms or fine structure terms is also
an expansion alpha squared.
So you may have-- the symmetry may allow it,
but only in connection with relativistic terms.
I'm not an expert on forbidden transitions,
but usually you have transitions which are multiply forbidden.
They're forbidden by spatial symmetry.
For instance, in the helium atom singular triplet
are forbidden by spin symmetry.
So if you have multiple layers of being forbidden,
then you get extremely weak transitions.
And one example is actually-- are
the transitions, the singlet-triplet transition, in helium
or the 1S to 2S transition in hydrogen.
They're not allowed by simply going
to the next order in the multipole expansion.
Yes.
All these equations, then we have
to do have to consider these even more highly
formulated transitions.
I'm not an expert on that.
I'm not sure if there is an atom which
is a relevant transition which is an extra high order transition.
I've not heard about that.
At least relevant examples which are the fundamental atoms
helium and hydrogen, for hydrogen, the 2S to 1S
transition, the leading order is the simultaneous emission
of tow photons.
So you have not just one photon.
You have two photons, which of course
requires in the perturbation expansion, intermediate step.
We discussed two photon transitions
at the end of this course.
So that case, it's not in higher order in the single photon
multipole expansion.
It becomes a multiphoton transaction.
So this is one relevant case.
And for helium triplet to singlet,
this involves relativistic physics.
And you really have to go-- actually,
I mentioned in the other class on helium.
I tried to look it up and I wanted
to show here you have to go to this order to get a transition.
But when I tried to look into the literature,
I couldn't find a clear answer.
Ultimately, it was a relativistic term
in the fully relativistic formulation
of the coupling of electromagnetic fields
to the atom.
So that can really-- I'm not sure if you
can put a label on it and would say,
this is this and this order term.
It may actually involve, and we know
that this happens to the Dirac equation,
that spin and spacial decrease become together
in the Dirac equation.
And maybe it's one of those terms.
Yes.
So what's the actual meaning of the interference between different Hamiltonians?
Like, you square the matrix elements
No, I meant actually the square of the matrix element.
If you have-- for instance, yeah, actually one--
I will talk about in the next lecture
when we discussed Fermi's golden rule.
The transition strengths in Fermi's golden rule
is proportional to the square of the matrix element.
But now you would ask the question, well,
could we have some interference between the two
different processes?
And I wanted to point out at this level that we don't.
Because one matrix element is imaginary.
The other one is real.
And if you take the complex matrix element between two
states and calculate the square, the square
of the complex matrix element is the square of the real part
plus the square of the imaginary part.
And there's no interference between the two.
So in other words, if you have an atom which has this weak
decay through M1 and a weak decay through quadrupole,
the two parts cannot destructively interfere
because one is real, one is imaginary.
They add up in quadratic.
That's what I wanted to say.